Saturday, August 06, 2011

Thomae's function

Here is the graph of this function with some points highlighted as plus symbols for better view.

This function has interesting property: it's continuous at all irrational numbers. It's easy to see this if you notice that for any positive ε there is finite number of points above the line y = ε. That means for any irrational number x0 you can always construct a δ-neighbourhood that doesn't contain any point from the area above the line y = ε.

To generate the data file with point coordinates I used Common Lisp program: